from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(166410, base_ring=CyclotomicField(1806))
M = H._module
chi = DirichletCharacter(H, M([1204,0,1793]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,166410))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(166410\) | |
| Conductor: | \(16641\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1806\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 16641.cx | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{903})$ |
| Fixed field: | Number field defined by a degree 1806 polynomial (not computed) |
First 31 of 504 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{166410}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{258}\right)\) | \(e\left(\frac{155}{903}\right)\) | \(e\left(\frac{255}{301}\right)\) | \(e\left(\frac{173}{903}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{470}{903}\right)\) | \(e\left(\frac{671}{1806}\right)\) | \(e\left(\frac{353}{903}\right)\) | \(e\left(\frac{221}{258}\right)\) | \(e\left(\frac{73}{903}\right)\) |
| \(\chi_{166410}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{258}\right)\) | \(e\left(\frac{500}{903}\right)\) | \(e\left(\frac{240}{301}\right)\) | \(e\left(\frac{92}{903}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{584}{903}\right)\) | \(e\left(\frac{1145}{1806}\right)\) | \(e\left(\frac{527}{903}\right)\) | \(e\left(\frac{251}{258}\right)\) | \(e\left(\frac{352}{903}\right)\) |
| \(\chi_{166410}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{258}\right)\) | \(e\left(\frac{893}{903}\right)\) | \(e\left(\frac{236}{301}\right)\) | \(e\left(\frac{251}{903}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{494}{903}\right)\) | \(e\left(\frac{1151}{1806}\right)\) | \(e\left(\frac{152}{903}\right)\) | \(e\left(\frac{173}{258}\right)\) | \(e\left(\frac{607}{903}\right)\) |
| \(\chi_{166410}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{258}\right)\) | \(e\left(\frac{401}{903}\right)\) | \(e\left(\frac{48}{301}\right)\) | \(e\left(\frac{500}{903}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{779}{903}\right)\) | \(e\left(\frac{1433}{1806}\right)\) | \(e\left(\frac{587}{903}\right)\) | \(e\left(\frac{119}{258}\right)\) | \(e\left(\frac{853}{903}\right)\) |
| \(\chi_{166410}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{258}\right)\) | \(e\left(\frac{139}{903}\right)\) | \(e\left(\frac{151}{301}\right)\) | \(e\left(\frac{394}{903}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{538}{903}\right)\) | \(e\left(\frac{1429}{1806}\right)\) | \(e\left(\frac{235}{903}\right)\) | \(e\left(\frac{85}{258}\right)\) | \(e\left(\frac{683}{903}\right)\) |
| \(\chi_{166410}(751,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{258}\right)\) | \(e\left(\frac{814}{903}\right)\) | \(e\left(\frac{174}{301}\right)\) | \(e\left(\frac{157}{903}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{604}{903}\right)\) | \(e\left(\frac{943}{1806}\right)\) | \(e\left(\frac{811}{903}\right)\) | \(e\left(\frac{211}{258}\right)\) | \(e\left(\frac{797}{903}\right)\) |
| \(\chi_{166410}(1381,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{258}\right)\) | \(e\left(\frac{403}{903}\right)\) | \(e\left(\frac{61}{301}\right)\) | \(e\left(\frac{811}{903}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{319}{903}\right)\) | \(e\left(\frac{661}{1806}\right)\) | \(e\left(\frac{376}{903}\right)\) | \(e\left(\frac{7}{258}\right)\) | \(e\left(\frac{551}{903}\right)\) |
| \(\chi_{166410}(2011,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{258}\right)\) | \(e\left(\frac{10}{903}\right)\) | \(e\left(\frac{65}{301}\right)\) | \(e\left(\frac{652}{903}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{409}{903}\right)\) | \(e\left(\frac{655}{1806}\right)\) | \(e\left(\frac{751}{903}\right)\) | \(e\left(\frac{85}{258}\right)\) | \(e\left(\frac{296}{903}\right)\) |
| \(\chi_{166410}(2221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{258}\right)\) | \(e\left(\frac{824}{903}\right)\) | \(e\left(\frac{239}{301}\right)\) | \(e\left(\frac{809}{903}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{110}{903}\right)\) | \(e\left(\frac{695}{1806}\right)\) | \(e\left(\frac{659}{903}\right)\) | \(e\left(\frac{167}{258}\right)\) | \(e\left(\frac{190}{903}\right)\) |
| \(\chi_{166410}(3211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{258}\right)\) | \(e\left(\frac{398}{903}\right)\) | \(e\left(\frac{179}{301}\right)\) | \(e\left(\frac{485}{903}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{566}{903}\right)\) | \(e\left(\frac{785}{1806}\right)\) | \(e\left(\frac{452}{903}\right)\) | \(e\left(\frac{29}{258}\right)\) | \(e\left(\frac{403}{903}\right)\) |
| \(\chi_{166410}(3271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{258}\right)\) | \(e\left(\frac{568}{903}\right)\) | \(e\left(\frac{80}{301}\right)\) | \(e\left(\frac{733}{903}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{295}{903}\right)\) | \(e\left(\frac{181}{1806}\right)\) | \(e\left(\frac{577}{903}\right)\) | \(e\left(\frac{55}{258}\right)\) | \(e\left(\frac{17}{903}\right)\) |
| \(\chi_{166410}(3631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{258}\right)\) | \(e\left(\frac{61}{903}\right)\) | \(e\left(\frac{246}{301}\right)\) | \(e\left(\frac{4}{903}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{418}{903}\right)\) | \(e\left(\frac{835}{1806}\right)\) | \(e\left(\frac{337}{903}\right)\) | \(e\left(\frac{67}{258}\right)\) | \(e\left(\frac{722}{903}\right)\) |
| \(\chi_{166410}(3931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{258}\right)\) | \(e\left(\frac{848}{903}\right)\) | \(e\left(\frac{94}{301}\right)\) | \(e\left(\frac{26}{903}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{903}\right)\) | \(e\left(\frac{461}{1806}\right)\) | \(e\left(\frac{836}{903}\right)\) | \(e\left(\frac{113}{258}\right)\) | \(e\left(\frac{178}{903}\right)\) |
| \(\chi_{166410}(4111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{258}\right)\) | \(e\left(\frac{563}{903}\right)\) | \(e\left(\frac{198}{301}\right)\) | \(e\left(\frac{407}{903}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{542}{903}\right)\) | \(e\left(\frac{305}{1806}\right)\) | \(e\left(\frac{653}{903}\right)\) | \(e\left(\frac{77}{258}\right)\) | \(e\left(\frac{772}{903}\right)\) |
| \(\chi_{166410}(4201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{258}\right)\) | \(e\left(\frac{767}{903}\right)\) | \(e\left(\frac{19}{301}\right)\) | \(e\left(\frac{524}{903}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{578}{903}\right)\) | \(e\left(\frac{1025}{1806}\right)\) | \(e\left(\frac{803}{903}\right)\) | \(e\left(\frac{5}{258}\right)\) | \(e\left(\frac{670}{903}\right)\) |
| \(\chi_{166410}(4291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{258}\right)\) | \(e\left(\frac{821}{903}\right)\) | \(e\left(\frac{69}{301}\right)\) | \(e\left(\frac{794}{903}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{800}{903}\right)\) | \(e\left(\frac{47}{1806}\right)\) | \(e\left(\frac{524}{903}\right)\) | \(e\left(\frac{77}{258}\right)\) | \(e\left(\frac{643}{903}\right)\) |
| \(\chi_{166410}(4441,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{258}\right)\) | \(e\left(\frac{727}{903}\right)\) | \(e\left(\frac{60}{301}\right)\) | \(e\left(\frac{625}{903}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{748}{903}\right)\) | \(e\left(\frac{211}{1806}\right)\) | \(e\left(\frac{508}{903}\right)\) | \(e\left(\frac{181}{258}\right)\) | \(e\left(\frac{389}{903}\right)\) |
| \(\chi_{166410}(4621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{258}\right)\) | \(e\left(\frac{625}{903}\right)\) | \(e\left(\frac{300}{301}\right)\) | \(e\left(\frac{115}{903}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{730}{903}\right)\) | \(e\left(\frac{1657}{1806}\right)\) | \(e\left(\frac{433}{903}\right)\) | \(e\left(\frac{217}{258}\right)\) | \(e\left(\frac{440}{903}\right)\) |
| \(\chi_{166410}(5251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{258}\right)\) | \(e\left(\frac{550}{903}\right)\) | \(e\left(\frac{264}{301}\right)\) | \(e\left(\frac{643}{903}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{823}{903}\right)\) | \(e\left(\frac{1711}{1806}\right)\) | \(e\left(\frac{670}{903}\right)\) | \(e\left(\frac{31}{258}\right)\) | \(e\left(\frac{26}{903}\right)\) |
| \(\chi_{166410}(5881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{258}\right)\) | \(e\left(\frac{388}{903}\right)\) | \(e\left(\frac{114}{301}\right)\) | \(e\left(\frac{736}{903}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{157}{903}\right)\) | \(e\left(\frac{1033}{1806}\right)\) | \(e\left(\frac{604}{903}\right)\) | \(e\left(\frac{73}{258}\right)\) | \(e\left(\frac{107}{903}\right)\) |
| \(\chi_{166410}(6091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{258}\right)\) | \(e\left(\frac{173}{903}\right)\) | \(e\left(\frac{71}{301}\right)\) | \(e\left(\frac{263}{903}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{845}{903}\right)\) | \(e\left(\frac{947}{1806}\right)\) | \(e\left(\frac{260}{903}\right)\) | \(e\left(\frac{245}{258}\right)\) | \(e\left(\frac{64}{903}\right)\) |
| \(\chi_{166410}(7081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{258}\right)\) | \(e\left(\frac{797}{903}\right)\) | \(e\left(\frac{214}{301}\right)\) | \(e\left(\frac{674}{903}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{902}{903}\right)\) | \(e\left(\frac{281}{1806}\right)\) | \(e\left(\frac{347}{903}\right)\) | \(e\left(\frac{131}{258}\right)\) | \(e\left(\frac{655}{903}\right)\) |
| \(\chi_{166410}(7141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{258}\right)\) | \(e\left(\frac{211}{903}\right)\) | \(e\left(\frac{17}{301}\right)\) | \(e\left(\frac{754}{903}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{232}{903}\right)\) | \(e\left(\frac{727}{1806}\right)\) | \(e\left(\frac{766}{903}\right)\) | \(e\left(\frac{181}{258}\right)\) | \(e\left(\frac{647}{903}\right)\) |
| \(\chi_{166410}(7501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{258}\right)\) | \(e\left(\frac{670}{903}\right)\) | \(e\left(\frac{141}{301}\right)\) | \(e\left(\frac{340}{903}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{313}{903}\right)\) | \(e\left(\frac{541}{1806}\right)\) | \(e\left(\frac{652}{903}\right)\) | \(e\left(\frac{19}{258}\right)\) | \(e\left(\frac{869}{903}\right)\) |
| \(\chi_{166410}(7801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{258}\right)\) | \(e\left(\frac{638}{903}\right)\) | \(e\left(\frac{234}{301}\right)\) | \(e\left(\frac{782}{903}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{449}{903}\right)\) | \(e\left(\frac{251}{1806}\right)\) | \(e\left(\frac{416}{903}\right)\) | \(e\left(\frac{5}{258}\right)\) | \(e\left(\frac{283}{903}\right)\) |
| \(\chi_{166410}(7981,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{258}\right)\) | \(e\left(\frac{626}{903}\right)\) | \(e\left(\frac{156}{301}\right)\) | \(e\left(\frac{722}{903}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{500}{903}\right)\) | \(e\left(\frac{1271}{1806}\right)\) | \(e\left(\frac{779}{903}\right)\) | \(e\left(\frac{161}{258}\right)\) | \(e\left(\frac{289}{903}\right)\) |
| \(\chi_{166410}(8071,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{258}\right)\) | \(e\left(\frac{641}{903}\right)\) | \(e\left(\frac{103}{301}\right)\) | \(e\left(\frac{797}{903}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{662}{903}\right)\) | \(e\left(\frac{899}{1806}\right)\) | \(e\left(\frac{551}{903}\right)\) | \(e\left(\frac{95}{258}\right)\) | \(e\left(\frac{733}{903}\right)\) |
| \(\chi_{166410}(8161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{258}\right)\) | \(e\left(\frac{338}{903}\right)\) | \(e\left(\frac{90}{301}\right)\) | \(e\left(\frac{185}{903}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{821}{903}\right)\) | \(e\left(\frac{467}{1806}\right)\) | \(e\left(\frac{461}{903}\right)\) | \(e\left(\frac{35}{258}\right)\) | \(e\left(\frac{433}{903}\right)\) |
| \(\chi_{166410}(8311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{258}\right)\) | \(e\left(\frac{412}{903}\right)\) | \(e\left(\frac{270}{301}\right)\) | \(e\left(\frac{856}{903}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{55}{903}\right)\) | \(e\left(\frac{799}{1806}\right)\) | \(e\left(\frac{781}{903}\right)\) | \(e\left(\frac{19}{258}\right)\) | \(e\left(\frac{95}{903}\right)\) |
| \(\chi_{166410}(8491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{258}\right)\) | \(e\left(\frac{436}{903}\right)\) | \(e\left(\frac{125}{301}\right)\) | \(e\left(\frac{73}{903}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{856}{903}\right)\) | \(e\left(\frac{565}{1806}\right)\) | \(e\left(\frac{55}{903}\right)\) | \(e\left(\frac{223}{258}\right)\) | \(e\left(\frac{83}{903}\right)\) |
| \(\chi_{166410}(9121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{258}\right)\) | \(e\left(\frac{697}{903}\right)\) | \(e\left(\frac{166}{301}\right)\) | \(e\left(\frac{475}{903}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{424}{903}\right)\) | \(e\left(\frac{955}{1806}\right)\) | \(e\left(\frac{61}{903}\right)\) | \(e\left(\frac{55}{258}\right)\) | \(e\left(\frac{404}{903}\right)\) |